Basics of Linear Regression

The analysis is based on the Gapminder codebook containing 15 socio-economic factors from 2007 drawn from 213 countries. Based on previous research review, I chose to study the correlation between the urban rate (a quantitative explanatory variable) and the female employ rate (a quantitative response variable) in the Gapminder dataset. The analysis was generated using SAS Studio and the GLM procedure to analyze the data within the framework of a general linear model. A standardized normal distribution of the explanatory variable was created using PROC STANDARD. The procedure computes a z-score by standardizing the urban rate to a mean of 0 and a standard deviation of 1. The variable is standardized to make it easier to interpret the results of the regression analysis

SAS CODE

%web_drop_table(WORK.GAPMINDER);

PROC IMPORT DATAFILE="/home/mst07221/gapminder.csv"
DBMS=CSV
OUT=WORK.GAPMINDER;
GETNAMES=YES;
RUN;

***********************************************************************
PROC STANDARD: A new explanatory variable is created from the current explanatory variable. 
PROC MEANS: the results are confirmed
**********************************************************************;
DATA gapminder2;
  SET WORK.GAPMINDER;
  zurbanrate = urbanrate;
RUN;

PROC STANDARD DATA=gapminder2 MEAN=0 STD=1 OUT=zgapminder2;
  VAR zurbanrate;
RUN;

PROC MEANS DATA=zgapminder2;
  VAR zurbanrate urbanrate femaleemployrate;
RUN;
  
***********************************************************************
BASIC LINEAR REGRESSION. Generate a scatter plot.
**********************************************************************;
PROC SGPLOT DATA=zgapminder2;
  reg x=zurbanrate y=femaleemployrate / lineattrs=(color=blue thickness=2);
  title "Scatterplot for the association Between Urban Rate and Female Employment Rate";
  yaxis label= "Female Employment Rate";
  xaxis label="Standardized Urban Rate";
RUN;
title;

***********************************************************************
The GLM procedure uses the method of least squares to fit general linear models;
The option 'solution' produces parameter estimates;  
**********************************************************************;
PROC GLM DATA=zgapminder2; 
model femaleemployrate=zurbanrate/solution;
RUN;

%web_open_table(WORK.GAPMINDER);

RESULTS

The MEANS Procedure

Variable	N	Mean	Std Dev	Minimum	Maximum
zurbanrate urbanrate femaleemployrate	203 203 178	7.328566E-16 56.7693596 47.5494381	1.0000000 23.8449326 14.6257425	-1.9446211 10.4000000 11.3000002	1.8129907 100.0000000 83.3000031

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The GLM Procedure

Number of Observations Read	213
Number of Observations Used	173

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The GLM Procedure

Dependent Variable: femaleemployrate

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	3446.74019	3446.74019	17.28	<.0001
Error	171	34100.73113	199.41948
Corrected Total	172	37547.47132

R-Square	Coeff Var	Root MSE	femaleemployrate Mean
0.091797	29.57241	14.12160	47.75260

Source	DF	Type I SS	Mean Square	F Value	Pr > F
zurbanrate	1	3446.740194	3446.740194	17.28	<.0001

Source	DF	Type III SS	Mean Square	F Value	Pr > F
zurbanrate	1	3446.740194	3446.740194	17.28	<.0001

Parameter	Estimate	Standard Error	t Value	Pr > |t|
Intercept	47.72703970	1.07366269	44.45	<.0001
zurbanrate	-4.58870848	1.10374814	-4.16	<.0001

SUMMARY OF FINDINGS

• Of the 213 observations available, 173 were used in the model.
• The overall F test is significant (F = 17.28, p < 0.0001 ). We can reject the NULL hypothesis and conclude that the urban rate is significantly associated with the female employ rate.
• The parameter estimates show a coefficient value of -4.59 and an intercept value of 47.73 (beta0 = 47.73, beta1 = -4.59).
• Therefore the best fit line equation is: femaleemployrate = 47.73 – 4.59 * urbanrate. This indicates a negative association between the two variables, also evident in the fit plot.
• The p-values for both the intercept and coefficient values are very small ( both p < 0.0001). This indicates there is indeed a straight-line relationship between femaleemployrate and urbanrate.
• The R-square value of 0.092 indicates that the proportion of variance in the response variable that can be attributed to the explanatory variable is only 9.2%. There may be other confounding variables that directly or inversely impact the relationship between the femaleemployrate and the urbanrate.